Linear Space Streaming Lower Bounds for Approximating CSPs

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on n variables taking values in {0,..., q - 1}, we prove that improving over the trivial approximability by a factor of q requires Omega(n) space even on instances with O( n) constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires O(n) space. The key technical core is an optimal, q-((k-1))-inapproximability for the Max k-LIN-mod q problem, which is the Max CSP problem where every constraint is given by a system of k - 1 linear equations mod q over k variables. Our work builds on and extends the breakthrough work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the MaxCut problem in graphs. MaxCut corresponds roughly to the case of Max k-LIN-mod q with k = q = 2. For general CSPs in the streaming setting, prior results only yielded Omega(root n) space bounds. In particular no linear space lower bound was known for an approximation factor less than 1/2 for any CSP. Extending the work of Kapralov and Krachun to Max k-LIN- mod q to k > 2 and q > 2 (while getting optimal hardness results) is the main technical contribution of this work. Each one of these extensions provides non-trivial technical challenges that we overcome in this work.